Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R4 with Z2-symmetry and integral of motion

Sajjad Bakrani; Jeroen S.W. Lamb; Dmitry Turaev

doi:10.1016/j.jde.2022.04.002

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R⁴ with Z₂-symmetry and integral of motion

Sajjad Bakrani, Jeroen S.W. Lamb, Dmitry Turaev

Research output: Contribution to journal › Article › peer-review

Abstract

We consider a Z₂-equivariant flow in R⁴ with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

Original language	English
Pages (from-to)	1-63
Number of pages	63
Journal	Journal of Differential Equations
Volume	327
DOIs	https://doi.org/10.1016/j.jde.2022.04.002
State	Published - 5 Aug 2022
Externally published	Yes

Keywords

Coupled Schrödinger equations
Homoclinic
Invariant manifold
Super-homoclinic

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jde.2022.04.002

Cite this

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title = "Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R4 with Z2-symmetry and integral of motion",

abstract = "We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schr{\"o}dinger equations is considered.",

keywords = "Coupled Schr{\"o}dinger equations, Homoclinic, Invariant manifold, Super-homoclinic",

author = "Sajjad Bakrani and Lamb, {Jeroen S.W.} and Dmitry Turaev",

note = "Funding Information: Acknowledgments. SB was supported by EU Marie Sklodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS), European Union ERC Advanced Grant of Sebastian Van Strien (339523 RGDD), and TUBITAK grant (No. 118C236). JSWL thanks the London Mathematical Laboratory, for support through its fellowship programme. DT was supported by Leverhulme Trust (RPG-2021-072), by the grants 19-11-00280 and 19-71-10048 of the Russian Science Foundation (RSF), by the Mathematical Center at the Lobachevsky University of Nizhny Novgorod, and by the grant 075-15-2019-1931 of Russian Ministry of Science and Higher Education. Publisher Copyright: {\textcopyright} 2022 The Authors",

year = "2022",

month = aug,

day = "5",

doi = "10.1016/j.jde.2022.04.002",

language = "English",

volume = "327",

pages = "1--63",

journal = "Journal of Differential Equations",

issn = "0022-0396",

publisher = "Academic Press Inc.",

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T2 - A case study in R4 with Z2-symmetry and integral of motion

AU - Bakrani, Sajjad

AU - Lamb, Jeroen S.W.

AU - Turaev, Dmitry

N1 - Funding Information: Acknowledgments. SB was supported by EU Marie Sklodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS), European Union ERC Advanced Grant of Sebastian Van Strien (339523 RGDD), and TUBITAK grant (No. 118C236). JSWL thanks the London Mathematical Laboratory, for support through its fellowship programme. DT was supported by Leverhulme Trust (RPG-2021-072), by the grants 19-11-00280 and 19-71-10048 of the Russian Science Foundation (RSF), by the Mathematical Center at the Lobachevsky University of Nizhny Novgorod, and by the grant 075-15-2019-1931 of Russian Ministry of Science and Higher Education. Publisher Copyright: © 2022 The Authors

PY - 2022/8/5

Y1 - 2022/8/5

N2 - We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

AB - We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

KW - Coupled Schrödinger equations

KW - Homoclinic

KW - Invariant manifold

KW - Super-homoclinic

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